[Retracted] Ordered-Theoretic Fixed Point Results in Fuzzy <i>b</i>-Metric Spaces with an Application

Javed, Khalil; Uddin, Fahim; Aydi, Hassen; Mukheimer, Aiman; Arshad, Muhammad

doi:https://doi.org/10.1155/2021/6663707

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Volume 2021 | Article ID 6663707 | https://doi.org/10.1155/2021/6663707

[Retracted] Ordered-Theoretic Fixed Point Results in Fuzzy b-Metric Spaces with an Application

Khalil Javed,¹Fahim Uddin,¹Hassen Aydi,^2,3,4Aiman Mukheimer,⁵and Muhammad Arshad¹

Academic Editor: Sami Ullah Khan

Received06 Dec 2020

Revised26 Dec 2020

Accepted02 Jan 2021

Published12 Feb 2021

Abstract

The aim of this manuscript is to initiate the study of the Banach contraction in R-fuzzy b-metric spaces and discuss some related fixed point results to ensure the existence and uniqueness of a fixed point. A nontrivial example is imparted to illustrate the feasibility of the proposed methods. Finally, to validate the superiority of the provided results, an application is presented to solve the first kind of a Fredholm-type integral equation.

1. Introduction and Preliminaries

Since the axiomatic interpretation of metric spaces and the inception of the Banach contraction principle, many authors have studied fixed point theory vividly. A number of results have been introduced, and metric fixed point has been generalized in different directions. In this connectedness, Bakhtin [1] and Czerwik [2] gave a generalization of a metric space and named it as a b-metric space. Zadeh [3] introduced the concept of fuzzy sets and generalized the concept of metric spaces and fuzzy sets and named them as fuzzy metric spaces, which became a point of interest for many authors [2, 4]. Nădăban [5] extended the concept of a fuzzy metric and introduced the notion of fuzzy b-metric spaces. For related works in this setting, refer to [6–9].

Recently, Baghani and Ramezani [10] tossed the concept of orthogonal sets and gave an extension of the Banach contraction principle. For more details, refer to [10–24].

In this article, we further aim to establish fixed point results in the setting of R-complete fuzzy b-metric spaces. We provide an example dealing with an R-fuzzy b-metric space, but it is not a fuzzy b-metric space. The presented results improve and generalize many results in the literature.

First, we recall some basic definitions and notions, which are essential for this work.

Definition 1 (see [11]). A binary operation : [0, 1] [0, 1] [0, 1] is referred to as a continuous t-norm if the following assumptions hold:(1)(2)(3)(4)If and , with , then Some fundamental examples of a t-norm are , and .

Definition 2 (see [12, 13]). A 3-tuple is said to be a fuzzy metric space if is an arbitrary set, is a continuous t-norm, and is a fuzzy set on meeting the following conditions for all :(B1)(B2) iff (B3)(B4)(B5) is continuous

Example 1 (see [12]). Let be a metric space with a continuous t-norm , and let be a fuzzy set defined on byThen, is called a standard fuzzy metric space.

Definition 3 (see [6]). A 4-tuple is said to be a fuzzy b-metric space if is an arbitrary set, is a continuous t-norm, and is a fuzzy set on meeting the following conditions for all and for a given real number :(B1)(B2) iff (B3)(B4)(B5) is continuous

Example 2 (see [7]). Let , where represents a real number. It is then simple to prove that is a fuzzy b-metric with . It should be noted that, for = 2, is not a fuzzy metric space.

Definition 4. Assume and is a binary relation. Suppose there exists such that or for all . Then, we say that is an R-set.

Example 3. (i)Let and define if ; then, by putting , is an R-set.(ii)Suppose is a set of scalar matrices of order with entries from natural numbers (i.e., , for all ). Define the relation R byThen, by taking , is an R-set.

Definition 5 (see [10]). Suppose that is an R-set. A sequence for all is said to be an R-sequence if or .

Definition 6. (see [14]).(a)A metric space is an R-metric space if is an R-set.(b)A mapping is R-continuous at if for each R-sequence for all in if , then . Furthermore, is R-continuous if is R-continuous at each .(c)A mapping is called R-preserving if , then for all.(d)An R-sequence in is said to be an R-Cauchy sequence if for every ε > 0, there exists an integer N such that d(, ) < ε for all n ≥ and m ≥ . It is clear that R or R.(e) is R-complete if every R-Cauchy sequence is convergent.

2. Main Results

We start this section with the introduction of R-fuzzy b-metric spaces.

Definition 7. Let and R be a reflexive binary relation on . Let be a continuous t-norm and be a fuzzy set on . Suppose that, for all and for all , with either ( or ), either ( or ), and either ( or ), the following conditions hold:(1)(2) if and only if (3)(4), where (5) is continuousThen, is called an R-fuzzy b-metric space with the coefficient .

Remark 1. In the above definition, the set is endowed with a reflexive binary relation R, and is a fuzzy set on satisfying (1)–(5) for those comparable elements with respect to the reflexive binary relation R. An R-fuzzy b-metric may not be a fuzzy b-metric.
The following simplest example shows that the R-fuzzy b-metric with does not need to be a fuzzy b-metric with .

Example 4. Let and . Define a binary relation such that iff . It is clear that is an R-fuzzy b-metric on with .
Note that for , and , the following condition does not hold:So, is not a fuzzy b-metric.

Definition 8. Let represent an R-fuzzy b-metric space.(a)A sequence for all is said to be an R-sequence if or .(b)A Cauchy sequence is said to be an R-Cauchy sequence if or .(c)A mapping is R-continuous at if for each R-sequence for all in with for all , then for all . Furthermore, is R-continuous if is R-continuous at each .(d)A mapping is called R-preserving if , then for all.(e)If each R-Cauchy sequence is convergent, then is R-complete.Motivated by the work of Baghani and Ramezani [10] and Hezarjaribi et al. [14], we introduce the concept of Banach contraction principle in the setting of R-fuzzy b-metric spaces.

Definition 9. Let be an R-fuzzy b-metric space. A map is an R-contraction if there exists such that, for every and with , we have

Theorem 1. Assume that is an R-complete fuzzy b-metric space. Let be an R-continuous, R-contraction, and R-preserving mapping. Thus, has a unique fixed point . Furthermore,

Proof. Since is an R-complete fuzzy b-metric space, there exists such thatThis yields that . Assume thatSince is R-preserving, is an R-sequence and is an R-contraction. Thus,for all and . Therefore, by applying the above expression, we deducefor all and τ > 0. Thus, from (9) and (B4), we haveHere, is an arbitrary positive integer. We know that for all and From (10), we getThen, is an R-Cauchy sequence. The hypothesis of R-completeness of the fuzzy b-metric space ensures that there exists such that as for all Since is an R-continuous mapping, one writes as . Hence,As , we get ; hence, .
To show the uniqueness of the fixed point for the mapping , assume that and are two fixed points of such that . We haveSince is R-preserving, we can writefor all . Using (4), we haveHence,So, ; hence, is the unique fixed point.

Corollary 1. Let be an R-complete fuzzy b-metric space. Let be an R-contraction and R-preserving. Also, if is an R-sequence with , then for all . Therefore, has a unique fixed point . Furthermore, , for all and .

Proof. The proof of this result moves along the same lines as in Theorem 1, that is, is an R-Cauchy sequence and converges to . Hence, for all . From (4), we haveAlso,Hence,As , we get , and so, . The rest of the proof is the same as in Theorem 1.

Corollary 2. Let be an R-complete fuzzy b-metric space and be an R-continuous and R-preserving mapping. Suppose that there exist and such thatThen, has a unique fixed point.

Corollary 3. Let be an R-complete fuzzy b-metric space and be an R-continuous and R-preserving mapping. Assume that there exist and such thatThen, has a unique fixed point.

Proof. The proof is a part of the next corollary.

Corollary 4. Let be an R-complete fuzzy b-metric space and be an R-continuous and R-preserving mapping. Assume that there exist and such thatThen, has a unique fixed point.

Proof. This corollary is a generalization of Theorem 2.5 in [8]. It is easy to prove this result by the help of Theorem 1 of this article and Theorem 2.5 of [8].

Example 5. Let . The relation on is defined as . Define the R-fuzzy b-metric given as in Example 4:with the t-norm . Let be an R-sequence in such that . Hence, converges to 1. Therefore, is an R-complete fuzzy b-metric space with .
Define byNote the following:(1)If and , then and(2)If and , then and (3)If and , then and In all cases, we have . Thus, is an R-preserving map.
Let be an arbitrary R-sequence in so that converges to . Now,As converges to , we have .
Now, we need to show that . For this purpose, there are some cases.(1)Take ; then,(2)Take ; then, As converges to , we have .(3)Now, take and ; then,As , we can easily see .
Hence, is R-continuous.
For each with , we have the following. Case (a) For , we have Case (b) For , , we haveHence, is an R-contraction. Hence, by Theorem 1, has a unique fixed point.

3. An Application to an Integral Equation

Within this part, we apply Theorem 1.

Let be the set of all continuous real-valued functions defined on .

Now, we consider the following Fredholm-type integral equation of first kind:where . Define M as in Example 4, that is,

Then, is an R-complete fuzzy b-metric space.

Theorem 2. Assume that for , , and . Also, consider . Let be(i)R-preserving(ii)R-contraction(iii)R-continuousThen, the Fredholm-type integral equation of first kind in equation (31) has a unique solution.

Proof. Define by(i)Define R as . We have , and we want to show that . We see that and belong to because is a self-map. So, observe that if then it must be . Hence, is R-preserving.(ii)Observe that the existence of a fixed point of is equivalent to the existence of a solution of Fredholm-type integral equation (31). Now, for all , we have Hence, is an R-contraction.(iii)Suppose is an R-sequence in such that converges to . Because is R-preserving, is an R-sequence for each . From (ii), we haveAs , for all it is clear thatHence, is R-continuous.
Now, assume that and are two fixed points of ; then, we haveThus, for all,Taking the limit as and using the fact , we get .
Consequently, all the conditions of Theorem 1 hold. The operator therefore has a unique fixed point.

4. Conclusion

Herein, we introduced the notion of R-fuzzy b-metric spaces and we proved some related fixed point results. Moreover, we presented some examples to illustrate the feasibility of the proposed methods and obtained results. We have also enriched this work with an application. Since our framework is more general than the class of fuzzy and fuzzy b-metric spaces, our results extend and generalize several existing ones in the literature.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and agreed to the published version of the manuscript.

Acknowledgments

The fourth author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group no. RG-DES-2017-01-17).

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Copyright © 2021 Khalil Javed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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